An important task in the reception of a digital signal is to detect when the beginning of a transmission arrives at the receiver in the presence of noise and channel distortion. This is particularly important in a random access communications system i.e. a system where transmission times are not known in advance, such as wireless LANs based on the IEEE 802.11 standard. The primary task in designing the detection process of such a system is to give a high probability of detecting an arriving signal within a specified time from the signal start, while giving a low probability of false alarms, indication of a signal when no signal is present, with noise or interfering signals.
An important class of communication systems are so-called “direct sequence spread spectrum” systems, where a transmitted sequence is scrambled by a “spreading code”, a higher rate sequence which causes the transmitted signal to occupy a wider bandwidth and thereby be more robust against frequency-selective fading, such as can be caused by multipath interference.
A subclass of such signals is used in the IEEE 802.11 and IEEE 802.11b standards. These standards define the use of a length 11 Barker sequence, defined as +1, −1, +1, +1, −1, +1, +1, +1, −1, −1, −1, where the left-most value, or “chip” is output first. This code is used to multiply the transmitted signal at a rate 11 times faster than the transmitted signal; for the case of the preamble in IEEE 802.11 DSSS/IEEE 802.11b, the transmitted signal during the preamble consists of one of two known pseudo-random sequences modulated using differential BPSK, i.e. the output carrier wave is multiplied either by +1 or −1.
In a practical design, the sampling rate at the transmitter and receiver is often a multiple of the chip rate, e.g. 22 MHz, in which case each chip value is output twice. For simplicity, and without loss of generality, the following discussion only deals with a sampling rate that is the same as the chip rate.
From the receiver's viewpoint, one of the main benefits of the Barker code is that it has a very low autocorrelation of either −1 or 0 except for lag zero, when the output is 11. In this context, the autocorrelation of a sampled signal x(n) with lag τ is defined as:
      Rxx    ⁡          (      τ      )        =            ∑      n        ⁢                  x        ⁡                  (          n          )                    ·                        x          *                ⁡                  (                      n            -            τ                    )                    
The property of low autocorrelation means that, at the receiver, the incoming signal can be correlated with the Barker sequence used at the transmitter. Signals modulated with the Barker sequence will therefore produce a strong peak at the output of the correlator, but other signals will not produce strong peaks.
During transmission of a single Barker sequence beginning at time m over an additive white gaussian channel, noise is added, leading to a received complex signal given by:y(n)=Aexp(iφ)x(n−m)+e(n)i.e. the received samples are scaled, phase-rotated versions of the original plus white gaussian noise.
When the Barker sequence is superimposed on a chain of BPSK symbols, we can use the linear nature of the autocorrelation to determine that the result will be the sum of the responses to each individual pulse. This will therefore produce a sequence of pulses at the end of each transmitted BPSK symbol, with the sign of the pulse depending on the sign of the transmitted symbol.
In a practical system, there are other degradations than simple Gaussian noise. One important source of degradation is frequency offset between the transmitting and receiving devices. In the 802.11b standard, this can be up to 25 ppm at each device, leading to a total possible mismatch of 50 ppm or 120 kHz. This causes the phase shift in the received signal to vary with time, and causes the Barker correlator result to have a changing phase and slightly reduced output signal amplitude. However, this amplitude reduction is small, e.g. −0.2 dB, at a frequency offset of 120 kHz.
Objects between a radio transmitter and receiver, which are reflective to radio energy, provide a range of different paths between transmitter and receiver other than the direct path, which may even not exist in some situations. Each of these paths has a different delay, and attenuates the signal by a different amount. The result at the receiver is that each received sample is the sum of a number of differently delayed copies of the signal with different phases and amplitudes, plus noise:
      y    ⁡          (      n      )        =                    ∑        m            ⁢                        A          m                ⁢                  exp          ⁡                      (                          i              ⁢                                                          ⁢                              ϕ                m                                      )                          ⁢                  x          ⁡                      (                          n              -              m                        )                                +          e      ⁡              (        n        )            
Again, superposition may be used to determine the correlator response. Each multipath ray m produces a distinct peak of amplitude 11Am surrounded by ripples with amplitude Am. This means that the correlator output can be used effectively to separate each multipath ray, since the ripples from adjacent multipaths are small in comparison.
The signal energy available at the receiver is spread out across all of the different multipaths. This means that, for a given signal energy, the peak correlator output is lower typically a “hump” rather than a sharp peak as in the case of non-dispersive channels.
The most straightforward solution to the Barker preamble detect problem is to base the detection decision on the difference between the Barker correlator output in the presence or absence of a Barker-modulated input signal. The mean output signal, per-symbol peak, in the presence of a Barker preamble is given by:
            c      ⁡              (        n        )                    bar      ⁢                          ⁢      ker        =            11      ⁢      A      ⁢                          ⁢              exp        ⁡                  (                      i            ⁢                                                  ⁢            ϕ                    )                      +                  ∑                  k          =          0                10            ⁢              e        ⁡                  (                      n            -            k                    )                    
In the absence of a signal the correlator output signal is given by:
            c      ⁡              (        n        )              noise    =            ∑              k        =        0            10        ⁢          e      ⁡              (                  n          -          k                )            
The main difficulty in this process is the unknown phase and amplitude of a given symbol. By taking the square value of the magnitude of the correlator output at a given instant, the phase component can be eliminated. The result is given by:
                                          c            ⁡                          (              n              )                                ⁢                                    c              *                        ⁡                          (              n              )                                      =                ⁢                              (                                          11                ⁢                A                ⁢                                                                  ⁢                                  exp                  ⁡                                      (                                          i                      ⁢                                                                                          ⁢                      ϕ                                        )                                                              +                                                ∑                                      k                    =                    0                                    10                                ⁢                                  e                  ⁡                                      (                                          n                      -                      k                                        )                                                                        )                    ⁢                      (                                          11                ⁢                A                ⁢                                                                  ⁢                                  exp                  ⁡                                      (                                                                  -                        i                                            ⁢                                                                                          ⁢                      ϕ                                        )                                                              +                                                ∑                                      k                    =                    0                                    10                                ⁢                                                      e                    *                                    ⁡                                      (                                          n                      -                      k                                        )                                                                        )                                                  =                ⁢                              121            ⁢                          A              2                                +                      22            ⁢            A            ⁢                                                  ⁢                          Re              ⁡                              (                                                      ∑                                          k                      =                      0                                        10                                    ⁢                                      e                    ⁡                                          (                                              n                        -                        k                                            )                                                                      )                                              +                                    ∑                              k                =                0                            10                        ⁢                                          e                ⁡                                  (                                      n                    -                    k                                    )                                            ⁢                                                e                  *                                ⁡                                  (                                      n                    -                    k                                    )                                                                        
It can be seen that this expression consists of three main components: a constant component proportional to the signal power, a component proportional to the noise power, and a cross-product which has zero mean but whose variance is proportional to the signal level and noise level. In the absence of a signal, the result is proportional to the noise power.
If this result is summed across S symbols, according to:
      d    ⁡          (      n      )        =            ∑              s        =        0            s        ⁢                  c        ⁡                  (                      n            -                          11              ⁢              s                                )                    ·                        c          *                ⁡                  (                      n            -                          11              ⁢              s                                )                    the variance of the result is reduced, leading to a lower probability for false alarm.
The result of this summation is still proportional to the amplitude of the received signal. One approach to setting a decision threshold in the presence of this variable amplitude is to measure the mean amplitude of the input signal, and use it to normalize the detection threshold.
The main weakness of the Barker correlation magnitude based solution is that the detection decision is based on the difference between the noise level and the signal plus noise level. When the signal to noise ratio is low, the difference between the expected outputs with and without a signal is low and the variance of the result is high, which leads to poor detector performance. In addition, this method does not resolve multipath.
The previous method was based only on the power of the Barker correlation output averaged over a number of symbols. However, for an 802.11b preamble, the signal is also BPSK modulated by one of two possible sequences. This additional information can be used to improve the gain of the detector. Instead of correlating on a symbol-by-symbol basis, the received signal can be correlated against the entire known transmitted sequence, and the magnitude of the result used for the decision. The result is given by:
      d    ⁡          (      n      )        =            ∑              s        =        0            s        ⁢                  b        ⁡                  (          s          )                    ⁢              c        ⁡                  (                      n            -                          11              ⁢              s                                )                    where b(s) is the BPSK signal value at each symbol s. This signal has a variance that is reduced according to the length of the summation, but in contrast to the previous design, the mean output in the presence of pure noise is zero. Since there are two possible sequences in IEEE 802.11b, two results must be calculated with the corresponding values of b(s), and the maximum chosen. This increases the false alarm probability.
The second method has, at first sight, a much better performance since the expected output in the absence of signal is zero. Indeed, this architecture is well known to be the mathematically optimum detector in the case of a pure white Gaussian channel. However, the presence of an unknown frequency offset means that the output of the receiver loses phase alignment with the transmitted BPSK sequence within a few symbols, preventing this method from being used. Even without a frequency offset, this method offers no resolution of multipath components.